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Mirrors > Home > MPE Home > Th. List > ressmplvsca | Structured version Visualization version GIF version |
Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
Ref | Expression |
---|---|
ressmpl.s | ⊢ 𝑆 = (𝐼 mPoly 𝑅) |
ressmpl.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressmpl.u | ⊢ 𝑈 = (𝐼 mPoly 𝐻) |
ressmpl.b | ⊢ 𝐵 = (Base‘𝑈) |
ressmpl.1 | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
ressmpl.2 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressmpl.p | ⊢ 𝑃 = (𝑆 ↾s 𝐵) |
Ref | Expression |
---|---|
ressmplvsca | ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressmpl.u | . . . . 5 ⊢ 𝑈 = (𝐼 mPoly 𝐻) | |
2 | eqid 2738 | . . . . 5 ⊢ (𝐼 mPwSer 𝐻) = (𝐼 mPwSer 𝐻) | |
3 | ressmpl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
4 | eqid 2738 | . . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝐻)) = (Base‘(𝐼 mPwSer 𝐻)) | |
5 | 1, 2, 3, 4 | mplbasss 21355 | . . . 4 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝐻)) |
6 | 5 | sseli 3939 | . . 3 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘(𝐼 mPwSer 𝐻))) |
7 | eqid 2738 | . . . 4 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
8 | ressmpl.h | . . . 4 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
9 | eqid 2738 | . . . 4 ⊢ ((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))) = ((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))) | |
10 | ressmpl.2 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
11 | 7, 8, 2, 4, 9, 10 | resspsrvsca 21339 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ (Base‘(𝐼 mPwSer 𝐻)))) → (𝑋( ·𝑠 ‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋( ·𝑠 ‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌)) |
12 | 6, 11 | sylanr2 682 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋( ·𝑠 ‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌)) |
13 | 3 | fvexi 6854 | . . . 4 ⊢ 𝐵 ∈ V |
14 | 1, 2, 3 | mplval2 21354 | . . . . 5 ⊢ 𝑈 = ((𝐼 mPwSer 𝐻) ↾s 𝐵) |
15 | eqid 2738 | . . . . 5 ⊢ ( ·𝑠 ‘(𝐼 mPwSer 𝐻)) = ( ·𝑠 ‘(𝐼 mPwSer 𝐻)) | |
16 | 14, 15 | ressvsca 17185 | . . . 4 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘(𝐼 mPwSer 𝐻)) = ( ·𝑠 ‘𝑈)) |
17 | 13, 16 | ax-mp 5 | . . 3 ⊢ ( ·𝑠 ‘(𝐼 mPwSer 𝐻)) = ( ·𝑠 ‘𝑈) |
18 | 17 | oveqi 7365 | . 2 ⊢ (𝑋( ·𝑠 ‘(𝐼 mPwSer 𝐻))𝑌) = (𝑋( ·𝑠 ‘𝑈)𝑌) |
19 | fvex 6853 | . . . . 5 ⊢ (Base‘𝑆) ∈ V | |
20 | ressmpl.s | . . . . . . 7 ⊢ 𝑆 = (𝐼 mPoly 𝑅) | |
21 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
22 | 20, 7, 21 | mplval2 21354 | . . . . . 6 ⊢ 𝑆 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑆)) |
23 | eqid 2738 | . . . . . 6 ⊢ ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) = ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) | |
24 | 22, 23 | ressvsca 17185 | . . . . 5 ⊢ ((Base‘𝑆) ∈ V → ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) = ( ·𝑠 ‘𝑆)) |
25 | 19, 24 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) = ( ·𝑠 ‘𝑆) |
26 | fvex 6853 | . . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝐻)) ∈ V | |
27 | 9, 23 | ressvsca 17185 | . . . . 5 ⊢ ((Base‘(𝐼 mPwSer 𝐻)) ∈ V → ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) = ( ·𝑠 ‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))) |
28 | 26, 27 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘(𝐼 mPwSer 𝑅)) = ( ·𝑠 ‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻)))) |
29 | ressmpl.p | . . . . . 6 ⊢ 𝑃 = (𝑆 ↾s 𝐵) | |
30 | eqid 2738 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
31 | 29, 30 | ressvsca 17185 | . . . . 5 ⊢ (𝐵 ∈ V → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃)) |
32 | 13, 31 | ax-mp 5 | . . . 4 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑃) |
33 | 25, 28, 32 | 3eqtr3i 2774 | . . 3 ⊢ ( ·𝑠 ‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻)))) = ( ·𝑠 ‘𝑃) |
34 | 33 | oveqi 7365 | . 2 ⊢ (𝑋( ·𝑠 ‘((𝐼 mPwSer 𝑅) ↾s (Base‘(𝐼 mPwSer 𝐻))))𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌) |
35 | 12, 18, 34 | 3eqtr3g 2801 | 1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝐵)) → (𝑋( ·𝑠 ‘𝑈)𝑌) = (𝑋( ·𝑠 ‘𝑃)𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 ↾s cress 17072 ·𝑠 cvsca 17097 SubRingcsubrg 20171 mPwSer cmps 21259 mPoly cmpl 21261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-1st 7914 df-2nd 7915 df-supp 8086 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-fsupp 9265 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-uz 12723 df-fz 13380 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-tset 17112 df-subg 18884 df-ring 19920 df-subrg 20173 df-psr 21264 df-mpl 21266 |
This theorem is referenced by: ressply1vsca 21555 |
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